Day 1: run intervals. About 1.5-2mi of volume right not including walking for rest
Day 2: Barbell/DB stuff in the gym (full body of compound lifts and simple strength work) and possibly a short conditioner at the end.
Day 3: Light day, maybe do some BW stuff at home. Handstand work, Ring work, Lunges/Pistols/BW squats. More of active rest.

I had been doing the intervals after the barbell day and noticed my lower back was something of a limiter and I was probably slower on the splits.

I still have some decent work in on the squats and pulls the day after the intervals though my hips and quads are not quite fresh and feel a bit tight. I not really pushing my strength atm as I want to bring up my conditioning level and lean out and just maintain strength. I also don't feel like training 3-4hrs/day in the gym and so have walked away from that.

However, that sort of mileage makes me think the hips, legs, and calves would be far more fatigued even if you are adapted to it.

I'd prioritize the running if that's your goal, but bare in mind the lower body strength will probably be hampered.